Sculpting a Volcano with the Lathe Tool

Level designers build round terrain, like volcanoes, craters and hills, by drawing one profile and spinning it around a vertical axis. To find how much rock that adds, wrap the shape in thin cylindrical shells and add them up: the shell method.

Drag to turn the solid. The amber region is what gets revolved.

How it works

Volumes by cylindrical shells

In one sentence

If the region under $y = h(x) \ge 0$ for $0 \le a \le x \le b$ is revolved about the $y$ -axis, the volume is $V = 2\pi\int_a^b x\,h(x)\,dx$ .

General formula

{\htmlData{sym=0}{V}} = 2\pi\int_0^{1.6} {\htmlData{sym=1}{x}}\,{\htmlData{sym=2}{h(x)}}\,{\htmlData{sym=3}{dx}}, \qquad {\htmlData{sym=2}{h(x)}} = 1.5\,e^{-({\htmlData{sym=1}{x}} - {\htmlData{sym=4}{c}})^2/0.22}

Each shell, cut and unrolled, is a thin slab whose length is its circumference $2\pi x$ . Outer shells count more because they're longer.

What the symbols mean

$V$
Volume
Cubic kilometers of rock in the volcano.
$x$
Shell radius
Distance from the volcano's center.
$h(x)$
Shell height
The ground height at distance x.
$dx$
Shell thickness
Width of each ring of ground.
$c$
Crater radius
Where the rim peaks; 0 gives a pointed summit.

The profile gives the ground height $h(x)$ at distance $x$ from the volcano's center.
A thin ring of ground at radius $x$ is a cylindrical shell: height $h(x)$ , thickness $dx$ , circumference $2\pi x$ .
Unrolled, a shell is a flat slab of volume $2\pi x \cdot h(x)\,dx$ .
Adding all the shells from the center out gives $V = 2\pi\int_a^b x\,h(x)\,dx$ .

Learn the math

Volumes by Cylindrical ShellsCalc I · Lesson 51, Applications of Integrals